The Anomalous Scaling Exponents of Turbulence in General Dimension from Random Geometry

Abstract

We propose an exact analytical formula for the anomalous scaling exponents of inertial range structure functions in incompressible fluid turbulence. The formula is a gravitational Knizhnik-Polyakov-Zamolodchikov (KPZ)-type relation, and is valid in any number of space dimensions. It incorporates intermittency by gravitationally dressing the Kolmogorov linear scaling via a coupling to a random geometry. The formula has one real parameter $\gamma$ that depends on the number of space dimensions. The scaling exponents satisfy the convexity inequality, and the supersonic bound constraint. They agree with the experimental and numerical data in two and three space dimensions, and with numerical data in four space dimensions. Intermittency increases with $\gamma$, and in the infinite $\gamma$ limit the scaling exponents approach the value one, as in Burgers turbulence. At large $n$ the $n$th order exponent scales as $\sqrt{n}$. We discuss the relation between fluid flows and black hole geometry that inspired our proposal.

18 pages, 3 figures; v3: additional clarifications, added references; v2: improved discussion, added one figure

References in corpus (5)

• Nonlinear Fluid Dynamics from Gravity
• The Incompressible Non-Relativistic Navier-Stokes Equation from Gravity
• Forced Fluid Dynamics from Gravity
• KPZ in one dimensional random geometry of multiplicative cascades
• The Incompressible Navier-Stokes Equations From Black Hole Membrane Dynamics

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• Scaling Relations in Two-Dimensional Relativistic Hydrodynamic Turbulence
• Stochastic gravity and turbulence
• On the fractal dimension of turbulent black holes
• On Scale Versus Conformal Symmetry in Turbulence
• Asymmetry of velocity increments in turbulence
• Neural Network Complexity of Chaos and Turbulence